Finding Appropriate z*-Values for Given Confidence Levels
The three practice questions here can help you achieve an understanding of the relationship between values of z and the confidence level needed for a margin of error. Use the following table to find the appropriate z*-value for the confidence levels given except where noted.
The Z-table and the preceding table are related but not the same. To see the connection, find the z*-value that you need for a 95% confidence interval by using the Z-table:
First off, if you look at the z*-table, you see that the number you need for z* for a 95% confidence interval is 1.96. However, when you look up 1.96 on the Z-table, you get a probability of 0.975. Why?
In a nutshell, the Z-table shows only the probability below a certain z-value, and you want the probability between two z-values, –z and z. If 95% of the values must lie between –z and z, you expand this idea to notice that a combined 5% of the values lie above z and below –z. So 2.5% of the values lie above z, and 2.5% of the values lie below –z.
To get the total area below this z-value, take the 95% between –z and z plus the 2.5% below –z, and you get 97.5%. That’s the z-value with 97.5% area below it. It’s also the number with 95% lying between two z-values, –z and z.
To avoid all these extra steps and headaches, the z*-table has already done this conversion for you. So when you look up 1.96, you automatically find 95% (not 97.5%).
What is the z*-value for a 99% confidence level?
The z*-table shows the answer: A 99% confidence level has a z*-value of 2.58.
What is the z*-value for a confidence level of 80%?
The z*-table shows the answer: An 80% confidence level has a z*-value of 1.28.
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